The Flex Divisor of a K3 Surface

Abstract

Abstract The flex divisor  $R_{\textrm flex}$ of a primitively polarized K3 surface $(X,L)$ is, generically, the set of all points $x\in X$ for which there exists a pencil $V\subset |L|$ whose base locus is $\{x\}$. We show that if $L^2=2d$ then $R_{\textrm flex}\in |n_dL|$ with $$ \begin{align*} &n_d= \frac{(2d)!(2d+1)!}{d!^2(d+1)!^2} =(2d+1)C(d)^2,\end{align*}$$where $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $F_{2d}$ of polarized K3 surfaces.